Every week to complement the video lectures, we will have a look at an example of a particular signal, that we call the signal of the day. Our aim is to show you cool, real-life applications of signal processing. These signals come from a large variety of fields. Environmental sciences, acoustics, image processing and more. Whatever your background is, there should be something for you. Moreover, the signal of the day, is always related to the theory. You're currently learning in the class, and thus, illustrates how the sometimes complex theory could be put to work. We hope you enjoy this Signal of the Day, as much as we had fun preparing them. Today, we are going to investigate the longest recorded signal of daily measurements. This signal consists of a series of daily mean temperature, recorded in the city of Vienna in Germany. These measurements were originally initiated, by one of the most famous German writers of all times, Johann Wolfgang von Goethe. Although Goethe is nowadays better known for his writings. He had a keen interest in sciences. In particular, meteorology. For example, he popularized the use of the so called Goethes barometer, which measures atmospheric pressure. He also started to consistently record the weather in his city Yena. And thus can be seen as one of the first digital signal processing experts. Today this heritage is preserved and extended by Professor Hyman at the Max Plank Institute, whom we would like to thank for providing us with the data. Let us first have a look at the signal. In the background you see the city of Yena. In the foreground you see a sequence of daily measurements. For convenience of representation, we have computed from the daily time series of observations. The Annual Mean Temperature. First, see how the signal really corresponds to our idea of a digital signal. It consists of a series of measurements, taken at regular intervals that we call samples. Moreover, each sample has a discreet amplitude, which is related to the precision of the thermometers that was used. These are the two characteristics of visual signals. Furthermore, there are periods in the course of history, where the signal was not regularly recorded. For example, in 1870 or 1945. We will simply ignore these missing points in our computation. More over, in the period form 1821 to 1834, the overall amplitude of the signal is somehow higher, which was corrected by a change of equipment in 1835. Therefore, in all our further processing, we will simply ignore that early period. We are going to apply a very simple processing method to this signal. Compute a moving average. At any time a small n, it is simply the average of the last capital N observations. Where capital N is the window of observation, over which the average is computed. The result is a smooth version of the original signal, as differences are averaged out by this operation. Let us have a look, at how this works in practice. Consider a smaller portion of the signal depicted here. We would like to compute the moving average in 1854. Let us take arbitrarily a window of N is equal to 10 observations, which are highlighted in black. To compute the moving average in 1854, we simply take the average of these ten samples. You can see the average is slightly smaller, than the actual value in 1854. In 1855, we make a new observation. We slide the window of observation, by removing the oldest observation in 1845. And, adding the most recent observation 1855, then we computer the the moving out region 1855, by taking the outreach samples or zs observation. Again we plot the resulting value in red. This process carries on, until we have reached the end of the signal. We can devise a recursive procedure to compute this moving average. Let us start with the definition of the moving average here, in the first equation. Is the sum from m equals to 0 to N minus 1x, n minus N. Is the entire sum divided by capital N to compute the average. Now we take out the first term of the sum here. That's the term for N is equal to 0. What is left is the sum of m is equal 1 to capital N minus 1. Now we do something that looks silly. We add xn minus N. And we subtract it right away. So the sum of these two is 0. So nothing has changed. But then we recognized that, adding this changes this sum into the sum m is equal to 1, to N of x, n minus m of course divided by N. And this entire sum here, is simply y of small m minus 1. And this is what we write here, so we have three terms. We have yn minus 1, we have this character here, this character, and we have a new way to write this recursive averaging. We can represent this procedure using a block diagram, so let's write the equation again, we said. Yn is equal to y, n minus 1 plus 1 over n, x minus n minus x small n minus capital N, okay? So this part here. We can implement like this. We take xn, we delay it by zn minus 1, that's a delay term, by capital N samples and we subtract it from xn. This is this part. And then we scale it by y over n. And finally, y ends the output. E is a previous output which is y of n minus 1 added to what just came out from the x and minus x and minus capital N. Okay, just on here. All right. Now, we had seen block diagrams like this in the carplus strong example, where we had actually similar operations going on. The same time, we are going to spend more time later in the class to study such block diagrams, so don't get too worried you say, look, it will be too complex at this point in time. Let us now let us apply this moving average to our signal, with a window of 25 years to reveal long term trends. We observe a clear increase in temperature over the course of time, especially in the second part of the 20th century. This is a perfect illustration of the effect of global warming. The only question is, where does global warming coming from? And this is a subject of hot, indeed, hot political debate. What I want to say here is that the moving average, is a very simple tool, but it's used all over. For example, in finance, to average out small variations in financial times series. But also in environmental sensing, like this temperature example, or in other applications of signal processing, to various areas of science and engineering.